Question

A unit vector perpendicular to the vectors \[ \hat i-\hat j \] and \[ \hat i+\hat j \] is:

• A. \(\hat k\)
• B. \(-\dfrac{\hat i+\hat j}{\sqrt2}\)
• C. \(\dfrac{\hat i-\hat j}{\sqrt2}\)
• D. \(\dfrac{\hat i+\hat j}{\sqrt2}\)

Hint:

A vector perpendicular to two vectors: \[ \vec a \text{ and } \vec b \] is given by: \[ \vec a\times\vec b \]

Answer 1

The Correct Option is A

Step 1: Let the given vectors be \[ \vec a=\hat i-\hat j \] and \[ \vec b=\hat i+\hat j \] A vector perpendicular to both vectors is obtained using: \[ \vec a\times\vec b \]
Step 2: Find the cross product \[ \vec a\times\vec b= \begin{vmatrix} \hat i & \hat j & \hat k \\ 1 & -1 & 0 \\ 1 & 1 & 0 \end{vmatrix} \] \[ = \hat k(1+1) \] \[ =2\hat k \]
Step 3: Find the unit vector Magnitude of: \[ 2\hat k \] is: \[ 2 \] Hence unit vector is: \[ \frac{2\hat k}{2} = \hat k \] Option analysis:
• Option (A): Correct
• Option (B): Incorrect
• Option (C): Incorrect
• Option (D): Incorrect Therefore: \[ \boxed{\text{(A)}} \]